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Generalizations of Polya's urn Problem
Fan Chung1, Shirin Handjani2, and Doug Jungreis2
1Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
fan@ucsd.edu
2Center for Communications Research, IDA, 4320 Westerra Court, San Diego, CA 92121, USA
shandjan@math.ucsd.edu, jungreis@ccrwest.org
Annals of Combinatorics 7 (2) p.141-153 June, 2003
AMS Subject Classification: 05D40, 60C05, 60G20, 68R10, 91C99
Abstract:
We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1-p, place the ball in an existing bin, such that the probability that the ball is placed in a bin is proportional to , where m is the number of balls in that bin. For p=0, the number of bins is fixed and finite, and the behavior of the process depends on whether is greater than, equal to, or less than 1. We survey the known results and give new proofs for all three cases. We then consider the case p>0. When =1, this is equivalent to the so-called preferential attachment scheme which leads to power law distribution for bin sizes. When >1, we prove that a single bin dominates, i.e., as the number of balls goes to infinity, the probability that any new ball either goes into that bin or creates a new bin converges to 1. When p>0 and <1, we show that under the assumption that certain limits exist, the fraction of bins having m balls shrinks exponentially as a function of m. We then discuss further generalizations and pose several open problems.
Keywords: balls and bins, power law, positive feedback, web models

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